Vanessa works out for $\frac{2}{3}$ of an hour every day. To keep her exercise routines interesting, she includes different types of exercises, such as squats and sit-ups, in each workout. If each type of exercise takes $\frac{2}{9}$ of an hour, how many different types of exercise can Vanessa do in each workout?
Solution: To find out how many types of exercise Vanessa could do in each workout, divide the total amount of exercise time ( $\frac{2}{3}$ of an hour) by the amount of time each exercise type takes ( $\frac{2}{9}$ of an hour). $ \dfrac{{\dfrac{2}{3} \text{ hour}}} {{\dfrac{2}{9} \text{ hour per exercise}}} = {\text{ number of exercises}} $ Dividing by a fraction is the same as multiplying by the reciprocal. The reciprocal of ${\dfrac{2}{9} \text{ hour per exercise}}$ is ${\dfrac{9}{2} \text{ exercises per hour}}$ $ {\dfrac{2}{3}\text{ hour}} \times {\dfrac{9}{2} \text{ exercises per hour}} = {\text{ number of exercises}} $ $ \dfrac{{2} \cdot {9}} {{3} \cdot {2}} = {\text{ number of exercises}} $ Reduce terms with common factors by dividing the $2$ in the numerator and the $2$ in the denominator by $2$ $ \dfrac{{\cancel{2}^{1}} \cdot {9}} {{3} \cdot {\cancel{2}^{1}}} = {\text{ number of exercises}} $ Reduce terms with common factors by dividing the $9$ in the numerator and the $3$ in the denominator by $3$ $ \dfrac{{1} \cdot {\cancel{9}^{3}}} {{\cancel{3}^{1}} \cdot {1}} = {\text{ number of exercises}} $ Simplify: $ \dfrac{{1} \cdot {3}} {{1} \cdot {1}} = {3} $ Vanessa can do 3 different types of exercise per workout.